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Optimal Multiple Trading Times Under the Exponential OU Model

with Transaction Costs∗

Tim Leung† Xin Li‡ Zheng Wang§

February 27, 2018

Abstract

This paper studies the timing of trades under mean-reverting price dynamics subject to fixedtransaction costs. We solve an optimal double stopping problem to determine the optimal timesto enter and subsequently exit the market, when prices are driven by an exponential Ornstein-Uhlenbeck process. In addition, we analyze a related optimal switching problem that involvesan infinite sequence of trades, and identify the conditions under which the double stopping andswitching problems admit the same optimal entry and/or exit timing strategies. Among ourresults, we find that the investor generally enters when the price is low, but may find it optimalto wait if the current price is sufficiently close to zero. In other words, the continuation (waiting)region for entry is disconnected. Numerical results are provided to illustrate the dependence oftiming strategies on model parameters and transaction costs.

Keywords: optimal double stopping, optimal switching, exponential OU process, transaction costs

JEL Classification: C41, G11, G12

Mathematics Subject Classification (2010): 60G40, 91G10, 62L15

1 Introduction

One important problem commonly faced by individual and institutional investors is to determinewhen to buy and sell an asset. As a potential investor observes the price process of an asset, she candecide to enter the market immediately or wait for a future opportunity. After completing the firsttrade, the investor will need to select the time to close the position. This motivates us to investigatethe optimal sequential timing of trades.

Naturally, the optimal sequence of trading times should depend on the price dynamics of the riskyasset. For instance, if the price process is a super/sub-martingale, then the investor, who seeks tomaximize the expected liquidation value, will either sell immediately or wait forever. Such a trivialtiming arises when the underlying price follows a geometric Brownian motion. Similar observationscan also be found in, among others, Shiryaev et al. (2008). On the other hand, it has been widelyobserved that many asset prices exhibit mean reversion, ranging from equities and commoditiesto currencies and volatility indices (see Metcalf and Hassett (1995), Bessembinder et al. (1995),Casassus and Collin-Dufresne (2005), and references therein). To incorporate mean-reversion forpositive price processes, one popular choice for pricing and investment applications is the exponential

∗We thank two anonymous referees for their thorough reviews of our paper and their helpful remarks.†IEOR Department, Columbia University, New York, NY 10027; email: tl2497@columbia.edu.‡IEOR Department, Columbia University, New York, NY 10027; email: xl2206@columbia.edu.§IEOR Department, Columbia University, New York, NY 10027; email: zw2192@columbia.edu.

1

Ornstein-Uhlenbeck (XOU) model, as proposed by Schwartz (1997) for commodity prices, due toits analytical tractability. It also serves as the building block of more sophisticated mean-revertingmodels.

In this paper, we study the optimal timing of trades under the XOU model subject to transactioncosts. We consider two different but related formulations. First, we consider the trading problemwith a single entry and single exit with a fixed cost incurred at each transaction. This leads us toanalyze an optimal double stopping problem. In the second formulation, the investor is assumed toenter and exit the market infinitely many times with transaction costs. This gives rise to an optimalswitching problem.

We analytically derive the non-trivial entry and exit timing strategies. Under both approaches,it is optimal to sell when the asset price is sufficiently high, though at different levels. As forentry timing, we find that, under some conditions, it is optimal for the investor not to enter themarket at all when facing the optimal switching problem. In this case for the investor who has along position, the optimal switching problem reduces into an optimal stopping problem, where theoptimal liquidation level is identical to that of the optimal double stopping problem. Otherwise,the optimal entry timing strategies for the double stopping and switching problem are describedby the underlying’s first passage time to an interval that lies above level zero. In other words, thecontinuation region for entry is disconnected of the form (0, A) ∪ (B,+∞), with critical price levelsA and B (see Theorems 3.2 and 3.4 below). This means that the investor generally enters when theprice is low, but may find it optimal to wait if the current price is too close to zero. We find that thisphenomenon is a distinct consequence due to fixed transaction costs under the XOU model. Indeed,when there is no fixed costs, even if there are proportional transaction costs (see Zhang and Zhang(2008)), the entry timing is simply characterized by a single price level.

A typical solution approach for optimal stopping problems driven by diffusion involves the analyt-ical and numerical studies of the associated free boundary problems or variational inequalities (VIs);see e.g. Bensoussan and Lions (1982), Øksendal (2003), and Sun (1992). This approach is very use-ful when the structure of the optimal buy/sell strategies are known. In contrast to the VI approach,we solve the double stopping problem by characterizing the value functions (for entry and exit) asthe smallest concave majorant of the corresponding reward functions (see Dayanik and Karatzas(2003) and references therein). This allows us to directly construct the value function, without apriori finding a candidate value function or imposing conditions on the stopping and continuationregions, such as whether they are connected or not. In other words, our method will derive thestructure of the stopping and continuation regions as an output. Moreover, the VI method becomesmore challenging when the form of the reward function does not possess some amenable properties,such as convexity, monotonicity, and positivity. This gives another reason for our probabilistic ap-proach since the reward function for the entry problem is neither convex/concave, nor monotone,nor always positive. Having solved the optimal double stopping problem, we determine the optimalstructures of the buy/sell/wait regions. We then apply this to infer a similar solution structure forthe optimal switching problem and verify using the variational inequalities.

In the literature, Zhang and Zhang (2008) investigate the optimal switching problem under theXOU price dynamics, with slippage (proportional transaction cost). As extension, Kong and Zhang(2010) allow for short selling so that the investor can enter the market with either long or shortposition, and close it out in the next trade. As a numerical approach, Song et al. (2009) discussa stochastic approximation scheme to compute the optimal buying and selling price levels by apriori assuming a buy-low-sell-high strategy under the XOU model. In contrast to these studies,we study both optimal double stopping and switching problems specifically under exponential OUwith fixed transaction costs. In particular, the optimal entry timing with fixed transaction costs ischaracteristically different from that with slippage.

Zervos et al. (2013) consider an optimal switching problem with fixed transaction costs under

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a class of time-homogeneous diffusions, including the GBM, mean-reverting CEV underlying, andother models. However, their results are not applicable to the exponential OU model as it violatesAssumption 4 of their paper (see also Remark 4.5 below). Indeed, their model assumptions restrictthe optimal entry region to be represented by a single critical threshold, whereas we show that inthe XOU model the optimal entry region is characterized by two positive price levels.

As for related applications of optimal double stopping, Leung and Li (2015) study the optimaltiming to trade an OU price spread with a stop-loss constraint. Karpowicz and Szajowski (2007)analyze the double stopping times for a risk process from the insurance company’s perspective.The problem of timing to buy/sell derivatives has also been studied in Leung and Ludkovski (2011)(European and American options) and Leung and Liu (2012) (credit derivatives). Menaldi et al.(1996) study an optimal starting-stopping problem for general Markov processes, and provide themathematical characterization of the value functions.

The rest of the paper is structured as follows. In Section 2, we formulate both the optimal doublestopping and optimal switching problems. Then, we present our analytical and numerical results inSection 3. The proofs of our main results are detailed in Section 4. Finally, the Appendix containsthe proofs for a number of lemmas.

2 Problem Overview

In the background, we fix a probability space (Ω,F ,P), where P is the historical probability measure.In this section, we provide an overview of our optimal double stopping and switching problems, whichwill involve an exponential Ornstein-Uhlenbeck (XOU) process. The XOU process (ξt)t≥0 is definedby

ξt = eXt , dXt = µ(θ −Xt) dt+ σ dBt. (2.1)

Here, X is an OU process driven by a standard Brownian motion B, with constant parametersµ, σ > 0, θ ∈ R. In other words, X is the log-price of the positive XOU process ξ.

2.1 Optimal Double Stopping Problems

Given a risky asset with an XOU price process, we first consider the optimal timing to sell. If theshare of the asset is sold at some time τ , then the investor will receive the value ξτ = eXτ and paya constant transaction cost cs > 0. Denote by F the filtration generated by B, and T the set ofall F-stopping times. To maximize the expected discounted value, the investor solves the optimalstopping problem

V (x) = supτ∈T

Ex

e−rτ (eXτ − cs)

, (2.2)

where r > 0 is the constant discount rate, and Ex· ≡ E·|X0 = x.The value function V (x) represents the expected liquidation value associated with ξ. On the

other hand, the current price plus the transaction cost constitute the total cost to enter the trade.Before even holding the risky asset, the investor can always choose the optimal timing to start thetrade, or not to enter at all. This leads us to analyze the entry timing inherent in the tradingproblem. Precisely, we solve

J(x) = supν∈T

Ex

e−rν(V (Xν)− eXν − cb)

, (2.3)

with the constant transaction cost cb > 0 incurred at the time of purchase. In other words, thetrader seeks to maximize the expected difference between the value function V (Xν) and the current

3

eXν , minus transaction cost cb. The value function J(x) represents the maximum expected valueof the investment opportunity in the price process ξ, with transaction costs cb and cs incurred,respectively, at entry and exit. For our analysis, the transaction costs cb and cs can be different. Tofacilitate presentation, we denote the functions

hs(x) = ex − cs and hb(x) = ex + cb. (2.4)

If it turns out that J(X0) ≤ 0 for some initial value X0, then the investor will not start to tradeX. It is important to identify the trivial cases under any given dynamics. Under the XOU model,since supx∈R(V (x)−hb(x)) ≤ 0 implies that J(x) ≤ 0 for x ∈ R, we shall therefore focus on the casewith

supx∈R

(V (x)− hb(x)) > 0,

and solve for the non-trivial optimal timing strategy.

2.2 Optimal Switching Problems

Under the optimal switching approach, the investor is assumed to commit to an infinite number oftrades. The sequential trading times are modeled by the stopping times ν1, τ1, ν2, τ2, · · · ∈ T suchthat

0 ≤ ν1 ≤ τ1 ≤ ν2 ≤ τ2 ≤ . . . .

A share of the risky asset is bought and sold, respectively, at times νi and τi, i ∈ N. The investor’soptimal timing to trade would depend on the initial position. Precisely, under the XOU model,if the investor starts with a zero position, then the first trading decision is when to buy and thecorresponding optimal switching problem is

J(x) = supΛ0

Ex

∞∑

n=1

[e−rτnhs(Xτn)− e−rνnhb(Xνn)]

, (2.5)

with the set of admissible stopping times Λ0 = (ν1, τ1, ν2, τ2, . . . ), and the reward functions hs andhb defined in (2.1). On the other hand, if the investor is initially holding a share of the asset, thenthe investor first determines when to sell and solves

V (x) = supΛ1

Ex

e−rτ1hs(Xτ1) +∞∑

n=2

[e−rτnhs(Xτn)− e−rνnhb(Xνn)]

, (2.6)

with Λ1 = (τ1, ν2, τ2, ν3, . . . ).In summary, the optimal double stopping and switching problems differ in the number of trades.

Observe that any strategy for the double stopping problems (2.1) and (2.1) are also candidatestrategies for the switching problems (2.2) and (2.2) respectively. Therefore, it follows that V (x) ≤V (x) and J(x) ≤ J(x). Our objective is to derive and compare the corresponding optimal timingstrategies under these two approaches.

3 Summary of Analytical Results

We first summarize our analytical results and illustrate the optimal trading strategies. The methodof solutions and their proofs will be discussed in Section 4. We begin with the optimal stopping

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problems (2.1) and (2.1) under the XOU model. Denote the infinitesimal generator of the OUprocess X in (2) by

L =σ2

2

d2

dx2+ µ(θ − x)

d

dx. (3.1)

Recall that the classical solutions of the differential equation

Lu(x) = ru(x),

for x ∈ R, are (see e.g. p.542 of Borodin and Salminen (2002) and Prop. 2.1 of Alili et al. (2005)):

F (x) ≡ F (x; r) :=

∫ ∞

0u

rµ−1e

√

2µ

σ2 (x−θ)u−u2

2 du, (3.2)

G(x) ≡ G(x; r) :=

∫ ∞

0u

rµ−1e

√

2µ

σ2 (θ−x)u−u2

2 du. (3.3)

Direct differentiation yields that F ′(x) > 0, F ′′(x) > 0, G′(x) < 0 and G′′(x) > 0. Hence, weobserve that both F (x) and G(x) are strictly positive and convex, and they are, respectively, strictlyincreasing and decreasing.

Define the first passage time of X to some level κ by τκ = inft ≥ 0 : Xt = κ. As is well known,F and G admit the probabilistic expressions (see Ito and McKean (1965) and Rogers and Williams(2000)):

Exe−rτκ =

F (x)F (κ) if x ≤ κ,G(x)G(κ) if x ≥ κ.

3.1 Optimal Double Stopping Problems

We now present the result for the optimal exit timing problem.

Theorem 3.1 The optimal liquidation problem (2.1) admits the solution

V (x) =

eb∗−cs

F (b∗) F (x) if x < b∗,

ex − cs if x ≥ b∗,(3.4)

where the optimal log-price level b∗ for liquidation is uniquely found from the equation

ebF (b) = (eb − cs)F′(b). (3.5)

The optimal liquidation time is given by

τ∗ = inf t ≥ 0 : Xt ≥ b∗ = inf t ≥ 0 : ξt ≥ eb∗

.

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Theorem 3.2 Under the XOU model, the optimal entry timing problem (2.1) admits the solution

J(x) =

PF (x) if x ∈ (−∞, a∗),

V (x)− (ex + cb) if x ∈ [a∗, d∗],

QG(x) if x ∈ (d∗,+∞),

(3.6)

with the constants

P =V (a∗)− (ea

∗+ cb)

F (a∗), Q =

V (d∗)− (ed∗+ cb)

G(d∗),

and the critical levels a∗ and d∗ satisfying, respectively,

F (a)(V ′(a)− ea) = F ′(a)(V (a)− (ea + cb)), (3.7)

G(d)(V ′(d)− ed) = G′(d)(V (d)− (ed + cb)). (3.8)

The optimal entry time is given by

νa∗,d∗ := inft ≥ 0 : Xt ∈ [a∗, d∗].

In summary, the investor should exit the market as soon as the price reaches the upper leveleb

∗. In contrast, the optimal entry timing is the first time that the XOU price ξ enters the interval

[ea∗, ed

∗]. In other words, it is optimal to wait if the current price ξt is too close to zero, i.e. if

ξt < ea∗. Moreover, the interval [ea

∗, ed

∗] is contained in (0, eb

∗), and thus, the continuation region

for market entry is disconnected.

3.2 Optimal Switching Problems

We now turn to the optimal switching problems defined in (2.2) and (2.2) under the XOU model.To facilitate the presentation, we denote

fs(x) := (µθ +1

2σ2 − r)− µx+ rcse

−x,

fb(x) := (µθ +1

2σ2 − r)− µx− rcbe

−x.

Applying the operator L (see (3)) to hs and hb (see (2.1)), it follows that (L−r)hs(x) = exfs(x) and(L − r)hb(x) = exfb(x). Therefore, fs (resp. fb) preserves the sign of (L − r)hs (resp. (L − r)hb).It can be shown that fs(x) = 0 has a unique root, denoted by xs. However,

fb(x) = 0 (3.9)

may have no root, a single root, or two distinct roots, denoted by xb1 and xb2, if they exist. Thefollowing observations will also be useful:

fs(x)

> 0 if x < xs,

< 0 if x > xs,and fb(x)

< 0 if x ∈ (−∞, xb1) ∪ (xb2,+∞),

> 0 if x ∈ (xb1, xb2).(3.10)

The optimal switching problems have two different sets of solutions depending on the problemdata.

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Theorem 3.3 The optimal switching problems (2.2) and (2.2) admit the solutions

J(x) = 0, for x ∈ R, and V (x) =

eb∗−cs

F (b∗) F (x) if x < b∗,

ex − cs if x ≥ b∗,(3.11)

where b∗ satisfies (3.1), if any of the following mutually exclusive conditions holds:

(i) There is no root or a single root to equation (3.2).

(ii) There are two distinct roots to (3.2). Also

∃ a∗ ∈ (xb1, xb2) such that F (a∗)ea∗

= F ′(a∗)(ea∗

+ cb), (3.12)

and

ea∗+ cb

F (a∗)≥eb

∗− cs

F (b∗). (3.13)

(iii) There are two distinct roots to (3.2) but (ii) does not hold.

In Theorem 3.3, J = 0 means that it is optimal not to enter the market at all. On the otherhand, if one starts with a unit of the underlying asset, the optimal switching problem reduces to aproblem of optimal single stopping. Indeed, the investor will never re-enter the market after exit.This is identical to the optimal liquidation problem (2.1) where there is only a single (exit) trade.The optimal strategy in this case is the same as V in (3.1) – it is optimal to exit the market as soonas the log-price X reaches the threshold b∗.

We also address the remaining case when none of the conditions in Theorem 3.3 hold. As weshow next, the optimal strategy will involve both entry and exit thresholds.

Theorem 3.4 If there are two distinct roots to (3.2), xb1 and xb2, and there exists a number a∗ ∈(xb1, xb2) satisfying (ii) such that

ea∗+ cb

F (a∗)<eb

∗− cs

F (b∗), (3.14)

then the optimal switching problems (2.2) and (2.2) admit the solutions

J(x) =

PF (x) if x ∈ (−∞, a∗),

KF (x)− (ex + cb) if x ∈ [a∗, d∗],

QG(x) if x ∈ (d∗,+∞),

(3.15)

V (x) =

KF (x) if x ∈ (−∞, b∗),

QG(x) + ex − cs if x ∈ [b∗,+∞),(3.16)

where a∗ satisfies (ii), and

P = K −ea

∗+ cb

F (a∗), K =

ed∗G(d∗)− (ed

∗+ cb)G

′(d∗)

F ′(d∗)G(d∗)− F (d∗)G′(d∗), Q =

ed∗F (d∗)− (ed

∗+ cb)F

′(d∗)

F ′(d∗)G(d∗)− F (d∗)G′(d∗).

There exist unique critical levels d∗ and b∗ which are found from the nonlinear system of equations:

edG(d) − (ed + cb)G′(d)

F ′(d)G(d) − F (d)G′(d)=ebG(b)− (eb − cs)G

′(b)

F ′(b)G(b) − F (b)G′(b), (3.17)

edF (d)− (ed + cb)F′(d)

F ′(d)G(d) − F (d)G′(d)=ebF (b)− (eb − cs)F

′(b)

F ′(b)G(b) − F (b)G′(b). (3.18)

Moreover, the critical levels are such that d∗ ∈ (xb1, xb2) and b∗ > xs.

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The optimal strategy in Theorem 3.4 is described by the stopping times Λ∗0 = (ν∗1 , τ

∗1 , ν

∗2 , τ

∗2 , . . . ),

and Λ∗1 = (τ∗1 , ν

∗2 , τ

∗2 , ν

∗3 , . . . ), with

ν∗1 = inft ≥ 0 : Xt ∈ [a∗, d∗],

τ∗i = inft ≥ ν∗i : Xt ≥ b∗, and ν∗i+1 = inft ≥ τ∗i : Xt ≤ d∗, for i ≥ 1.

In other words, it is optimal to buy if the price is within [ea∗, ed

∗] and then sell when the price ξ

reaches eb∗. The structure of the buy/sell regions is similar to that in the double stopping case (see

Theorems 3.1 and 3.2). In particular, a∗ is the same as a∗ in Theorem 3.2 since the equations (3.2)and (ii) are equivalent. The level a∗ is only relevant to the first purchase. Mathematically, a∗ isdetermined separately from d∗ and b∗. If we start with a zero position, then it is optimal to enterif the price ξ lies in the interval [ea

∗, ed

∗]. However, on all subsequent trades, we enter as soon as

the price hits ed∗from above (after exiting at eb

∗previously). Hence, the lower level a∗ becomes

irrelevant after the first entry.Note that the conditions that differentiate Theorems 3.3 and 3.4 are exhaustive and mutually

exclusive. If the conditions in Theorem 3.3 are violated, then the conditions in Theorem 3.4 musthold. In particular, condition (ii) in Theorem 3.3 holds if and only if

∣

∣

∣

∣

∫ xb1

−∞Ψ(x)exfb(x)dx

∣

∣

∣

∣

<

∫ xb2

xb1

Ψ(x)exfb(x)dx, (3.19)

where

Ψ(x) =2F (x)

σ2W(x), and W(x) = F ′(x)G(x) − F (x)G′(x) > 0. (3.20)

Inequality (3.2) can be numerically verified given the model inputs.

3.3 Numerical Examples

We numerically implement Theorems 3.1, 3.2, and 3.4, and illustrate the associated entry/exitthresholds. In Figure 1 (left), the optimal entry levels d∗ and d∗ rise, respectively, from 0.7425 to0.7912 and from 0.8310 to 0.8850, as the speed of mean reversion µ increases from 0.5 to 1. On theother hand, the critical exit levels b∗ and b∗ remain relatively flat over µ. As for the critical lowerlevel a∗ from the optimal double stopping problem, Figure 1 (right) shows that it is decreasing inµ. The same pattern holds for the optimal switching problem since the critical lower level a∗ isidentical to a∗, as noted above.

We now look at the impact of transaction cost in Figure 2. On the left panel, we observe thatas the transaction cost cb increases, the gap between the optimal switching entry and exit levels, d∗

and b∗, widens. This means that it is optimal to delay both entry and exit. Intuitively, to counterthe fall in profit margin due to an increase in transaction cost, it is necessary to buy at a lower priceand sell at a higher price to seek a wider spread. In comparison, the exit level b∗ from the doublestopping problem is known analytically to be independent of the entry cost, so it stays constant ascb increases in the figure. In contrast, the entry level d∗, however, decreases as cb increases but muchless significantly than d∗. Figure 2 (right) shows that a∗, which is the same for both the optimaldouble stopping and switching problems, increases monotonically with cb.

In both Figures 1 and 2, we can see that the interval of the entry and exit levels, (d∗, b∗), asso-ciated with the optimal switching problem lies within the corresponding interval (d∗, b∗) from theoptimal double stopping problem. Intuitively, with the intention to enter the market again uponcompleting the current trade, the trader is more willing to enter/exit earlier, meaning a narrowed

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0.5 0.6 0.7 0.8 0.9 1

0.7

0.8

0.9

1

1.1

1.2

µ

d∗

b∗

d∗

b∗

0.5 0.6 0.7 0.8 0.9 1−9.4

−9.2

−9

−8.8

−8.6

−8.4

µ

a∗

Figure 1: (Left) The optimal entry and exit levels vs speed of mean reversion µ. Parameters: σ = 0.2, θ = 1,r = 0.05, cs = 0.02, cb = 0.02. (Right) The critical lower level of entry region a∗ decreases monotonically from-8.4452 to -9.2258 as µ increases from 0.5 to 1. Parameters: σ = 0.2, θ = 1, r = 0.05, cs = 0.02, cb = 0.02.

0.02 0.04 0.06 0.08 0.10.7

0.8

0.9

1

1.1

1.2

cb

d∗

b∗

d∗

b∗

0.02 0.04 0.06 0.08 0.1−9.5

−9

−8.5

−8

−7.5

−7

−6.5

cb

a∗

Figure 2: (Left) The optimal entry and exit levels vs transaction cost cb. Parameters: µ = 0.6, σ = 0.2,θ = 1, r = 0.05, cs = 0.02. (Right) The critical lower level of entry region a∗ increases monotonically from-9.4228 to -6.8305 as cb increases from 0.01 to 0.1. Parameters: µ = 0.6, σ = 0.2, θ = 1, r = 0.05, cs = 0.02.

waiting region.

Figure 3 shows a simulated path and the associated entry/exit levels. As the path starts at

ξ0 = 2.6011 > ed∗> ed

∗, the investor waits to enter until the path reaches the lower level ed

∗

(double stopping) or ed∗(switching) according to Theorems 3.2 and 3.4. After entry, the investor

exits at the optimal level eb∗(double stopping) or eb

∗(switching). The optimal switching thresholds

imply that the investor first enters the market on day 188 where the underlying asset price is 2.3847.In contrast, the optimal double stopping timing yields a later entry on day 845 when the price firstreaches ed

∗= 2.1754. As for the exit timing, under the optimal switching setting, the investor

exits the market earlier on day 268 at the price eb∗= 2.8323. The double stopping timing is much

later on day 1160 when the price reaches eb∗= 3.0988. In addition, under the optimal switching

problem, the investor executes more trades within the same time span. As seen in the figure, theinvestor would have completed two ‘round-trip’ (buy-and-sell) trades in the market before the doublestopping investor liquidates for the first time.

9

0 200 400 600 800 1000 12001

1.5

2

2.5

3

3.5

4

4.5

Days

exp(b∗)

exp(b∗)

exp(d∗)exp(d∗)

Figure 3: A sample exponential OU path, along with entry and exit levels. Under the double stoppingsetting, the investor enters at νd∗ = inft ≥ 0 : ξt ≤ ed

∗

= 2.1754 with d∗ = 0.7772, and exit at τb∗ = inft ≥νd∗ : ξt ≥ eb

∗

= 3.0988 with b∗ = 1.1310. The optimal switching investor enters at νd∗

= inft ≥ 0 : ξt ≤

ed∗

= 2.3888 with d∗ = 0.8708, and exit at τb∗

= inft ≥ νd∗ : ξt ≥ eb

∗

= 2.8323 with b∗ = 1.0411. The

critical lower threshold of entry region is ea∗

= 1.264 · 10−4 with a∗ = −8.9760 (not shown in this figure).Parameters: µ = 0.8, σ = 0.2, θ = 1, r = 0.05, cs = 0.02, cb = 0.02.

4 Methods of Solution and Proofs

We now provide detailed proofs for our analytical results in Section 3 beginning with Theorems 3.1and 3.2 for the optimal double stopping problems.

4.1 Optimal Double Stopping Problems

Starting at any x ∈ R, we denote by τa ∧ τb the exit time from an interval [a, b] with −∞ ≤ a ≤x ≤ b ≤ +∞. If a = −∞, then we have τa = +∞ a.s. In effect, this removes the lower exit level.Similarly, it is possible that b = +∞, and τb = +∞ a.s. Consequently, by considering interval-typestrategies, we also include the class of stopping strategies of reaching a single level (as in Theorem3.1 above).

Now, let us introduce the transformation

ψ(x) :=F

G(x), (4.1)

and define za = ψ(a), zb = ψ(b). With the reward function hs(x), and using (4.1), we compute the

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corresponding expected discounted reward:

Exe−r(τa∧τb)hs(Xτa∧τb) = hs(a)Exe

−r(τa∧τb)11τa<τb+ hs(b)Exe−r(τa∧τb)11τa>τb (4.2)

= hs(a)F (x)G(b) − F (b)G(x)

F (a)G(b) − F (b)G(a)+ hs(b)

F (a)G(x) − F (x)G(a)

F (a)G(b) − F (b)G(a)

= G(x)

[

hs(a)

G(a)

ψ(b)− ψ(x)

ψ(b)− ψ(a)+hs(b)

G(b)

ψ(x)− ψ(a)

ψ(b) − ψ(a)

]

= G(ψ−1(z))

[

H(za)zb − z

zb − za+H(zb)

z − zazb − za

]

, (4.3)

where

H(z) :=

hsG

ψ−1(z) if z > 0,

limx→−∞

(hs(x))+

G(x) if z = 0.(4.4)

The last equality (4.1) transforms the problem from x coordinate to z = ψ(x) coordinate (see (4.1)).In turn, the candidate optimal exit interval [a∗, b∗] is determined by maximizing the expectation

in (4.1). This is equivalent to maximizing (4.1) over za and zb in the transformed problem. As aresult, for every z ≥ 0, we have

W (z) := supza,zb:za≤z≤zb

H(za)zb − z

zb − za+H(zb)

z − zazb − za

, (4.5)

which is the smallest concave majorant of H. Applying (4.1) to (4.1), we can express the maximalexpected discounted reward as

G(x)W (ψ(x)) = supa,b:a≤x≤b

Exe−r(τa∧τb)hs(Xτa∧τb).

Now, it remains to prove the optimality of the proposed stopping strategy. This also providesan analytic expression for the value function.

Theorem 4.1 Under the XOU model (2), the value function V (x) defined in (2.1) is given by

V (x) = G(x)W (ψ(x)),

where G, ψ and W are defined in (3), (4.1) and (4.1), respectively.

The proof is similar to that of Theorem 3.2 in Leung and Li (2015), and is thus omitted.By Theorem 4.1, it is sufficient to consider interval-type strategies for the optimal liquidation

problem under the XOU model. Note that the optimal levels (a∗, b∗) can depend on the initialvalue x, and they may coincide or take values −∞ or +∞. As such, the structure of the stoppingand continuation regions can potentially be characterized by multiple intervals, leading to discon-nected continuation regions (see Theorem 3.2 above). In order to determine the optimal exit timingstrategies and solve for V , the major challenge lies in analyzing the functions H and W .

4.1.1 Optimal Exit Timing

In preparation for the next result, we apply (4.1) and (3)-(3) to the definition of H in (4.1), andsummarize the crucial properties of H.

Lemma 4.2 The function H is continuous on [0,+∞), twice differentiable on (0,+∞) and pos-sesses the following properties:

11

(i) H(0) = 0, and

H(z)

< 0 if z ∈ (0, ψ(ln cs)),

> 0 if z ∈ (ψ(ln cs),+∞).

(ii) H(z) is strictly increasing for z ∈ (ψ(ln cs),+∞), and H ′(z) → 0 as z → +∞.

(iii)

H(z) is

convex if z ∈ (0, ψ(xs)],

concave if z ∈ [ψ(xs),+∞).

Based on Lemma 4.2, We sketch H in Figure 4. Using the properties of H, we now solve for theoptimal exit timing.

0 z

H

W

z∗ = ψ(b∗)ψ(ln cs)

ψ(xs)

Figure 4: Sketches of H and W . By Lemma 4.2, H is convex on the left of ψ(xs) and concave on the right.

The smallest concave majorant W is a straight line tangent to H at z∗ on [0, z∗), and coincides with H on

[z∗,+∞).

Proof of Theorem 3.1 We look for the value function of the form: V (x) = G(x)W (ψ(x)), whereW is the smallest concave majorant of H. By Lemma 4.2, we observe that H is concave over[ψ(xs),+∞), strictly positive over (ψ(ln cs),+∞), and H ′(z) → 0 as z → +∞. Therefore, thereexists a unique number z∗ > ψ(xs) ∨ ψ(ln cs) such that

H(z∗)

z∗= H ′(z∗). (4.6)

In turn, the smallest concave majorant of H is given by

W (z) =

zH(z∗)z∗

if z ∈ [0, z∗),

H(z) if z ∈ [z∗,+∞).

Substituting b∗ = ψ−1(z∗) into (4.1.1), we have

H(z∗)

z∗=H(ψ(b∗))

ψ(b∗)=eb

∗− cs

F (b∗),

12

and

H ′(z∗) =eψ

−1(z∗)G(ψ−1(z∗))− (eψ−1(z∗) − cs)G

′(ψ−1(z∗))

F ′(ψ−1(z∗))G(ψ−1(z∗))− F (ψ−1(z∗))G′(ψ−1(z∗))

=eb

∗G(b∗)− (eb

∗− cs)G

′(b∗)

F ′(b∗)G(b∗)− F (b∗)G′(b∗).

Equivalently, we can express (4.1.1) in terms of b∗:

eb∗− cs

F (b∗)=eb

∗G(b∗)− (eb

∗− cs)G

′(b∗)

F ′(b∗)G(b∗)− F (b∗)G′(b∗),

which is equivalent to (3.1) after simplification. As a result, we have

W (ψ(x)) =

ψ(x)H(z∗)z∗

= F (x)G(x)

eb∗−cs

F (b∗) if x ∈ (−∞, b∗),

H(ψ(x)) = ex−csG(x) if x ∈ [b∗,+∞).

In turn, the value function V (x) = G(x)W (ψ(x)) is given by (3.1).

4.1.2 Optimal Entry Timing

We can directly follow the arguments that yield Theorem 4.1, but with the reward as h(x) =V (x)− hb(x) = V (x)− (ex + cb) and define H analogous to H:

H(z) :=

hG ψ−1(z) if z > 0,

limx→−∞

(h(x))+

G(x) if z = 0.

We will look for the value function with the form: J(x) = G(x)W (ψ(x)), where W is the smallestconcave majorant of H. The properties of H is given in the next lemma.

Lemma 4.3 The function H is continuous on [0,+∞), differentiable on (0,+∞), and twice differ-entiable on (0, ψ(b∗)) ∪ (ψ(b∗),+∞), and possesses the following properties:

(i) H(0) = 0, and there exists some b < b∗ such that H(z) < 0 for z ∈ (0, ψ(b)) ∪ [ψ(b∗),+∞).

(ii) H(z) is strictly decreasing for z ∈ [ψ(b∗),+∞).

(iii) Define the constant

x∗ = θ +σ2

2µ−r

µ− 1.

There exist some constants xb1 and xb2, with −∞ < xb1 < x∗ < xb2 < xs, that solve fb(x) = 0,such that

H(z) is

convex if z ∈ (0, ψ(xb1)) ∪ (ψ(xb2),+∞)

concave if z ∈ (ψ(xb1), ψ(xb2)),

and z1 := argmaxz∈[0,+∞) H(z) ∈ (ψ(xb1), ψ(xb2)).

Figure 5 gives a sketch of H according to Lemma 4.3, and illustrate the corresponding smallestconcave majorant W .

13

0 z

H

W

z0 = ψ(a∗)

z1 = ψ(d∗)

ψ(b)ψ(b∗)

Figure 5: Sketches of H and W . The smallest concave majorant W is a straight line tangent to H at z0 on[0, z0), coincides with H on [z0, z1], and is equal to H(z1) on (z1,+∞).

Proof of Theorem 3.2 As in Lemma 4.3 and Figure 5, by the definition of the maximizer of H,z1 satisfies the equation

H′

(z1) = 0. (4.7)

Also there exists a unique number z0 ∈ (xb1, z1) such that

H(z0)

z0= H

′

(z0). (4.8)

Using (4.1.2), (4.1.2) and Figure 5, W is a straight line tangent to H at z0 on [0, z0), coincideswith H on [z0, z1], and is equal to H(z1) on (z1,+∞). As a result,

W (z) =

zH′(z0) if z ∈ [0, z0),

H(y) if z ∈ [z0, z1],

H(z1) if z ∈ (z1,+∞).

Substituting a∗ = ψ−1(z0) into (4.1.2), we have

H(z0)

z0=V (a∗)− (ea

∗+ cb)

F (a∗),

and

H′

(z0) =G(a∗)(V ′(a∗)− ea

∗)−G′(a∗)(V (a∗)− (ea

∗+ cb))

F ′(a∗)G(a∗)− F (a∗)G′(a∗).

Equivalently, we can express condition (4.1.2) in terms of a∗:

V (a∗)− (ea∗+ cb)

F (a∗)=G(a∗)(V ′(a∗)− ea

∗)−G′(a∗)(V (a∗)− (ea

∗+ cb))

F ′(a∗)G(a∗)− F (a∗)G′(a∗),

which is equivalent to (3.2) after simplification. Also, we can express H′(z0) in terms of a∗:

H′

(z0) =H(z0)

z0=V (a∗)− (ea

∗+ cb)

F (a∗)= P.

In addition, substituting d∗ = ψ−1(z1) into (4.1.2), we have

G(d∗)(V ′(d∗)− ed∗)−G′(d∗)(V (d∗)− (ed

∗+ cb))

F ′(d∗)G(d∗)− F (d∗)G′(d∗)= 0,

14

which can be further simplified to (3.2). Furthermore, H(z1) can be written in terms of d∗:

H(z1) =V (d∗)− (ed

∗+ cb)

G(d∗)= Q.

By direct substitution of the expressions for W and the associated functions, we obtain the valuefunction in (3.2).

4.2 Optimal Switching Problems

Using the results derived in previous sections, we can infer the structure of the buy and sell regions ofthe switching problem and then proceed to verify its optimality. In this section, we provide detailedproofs for Theorems 3.3 and 3.4.

Proof of Theorem 3.3 (Part 1) First, with hs(x) = ex − cs, we differentiate to get

(

hsF

)′

(x) =(ex − cs)F

′(x)− exF (x)

F 2(x). (4.9)

On the other hand, by Ito’s lemma, we have

hs(x) = Exe−rths(Xt) − Ex

∫ t

0e−ru(L − r)hs(Xu)du

.

Note that

Exe−rths(Xt) = e−rt

(

e(x−θ)e−µt+θ+σ2

4µ(1−e−2µt)

− cs

)

→ 0 as t→ +∞.

This implies that

hs(x) = −Ex

∫ +∞

0e−ru(L − r)hs(Xu)du

= −G(x)

∫ x

−∞Ψ(s)(L − r)hs(s)ds− F (x)

∫ +∞

x

Φ(s)(L − r)hs(s)ds, (4.10)

where Ψ is defined in (3.2) and

Φ(x) :=2G(x)

σ2W(x).

The last line follows from Theorem 50.7 in Rogers and Williams (2000, p. 293). Dividing both sidesby F (x) and differentiating the RHS of (4.2), we obtain

(

hsF

)′

(x) = −

(

G

F

)′

(x)

∫ x

−∞Ψ(s)(L − r)hs(s)ds−

G

F(x)Ψ(x)(L − r)hs(x)− Φ(x)(L − r)hs(x)

=W(x)

F 2(x)

∫ x

−∞Ψ(s)(L − r)hs(s)ds =

W(x)

F 2(x)q(x),

where

q(x) :=

∫ x

−∞Ψ(s)(L − r)hs(s)ds.

15

Since W(x), F (x) > 0, we deduce that(

hsF

)′(x) = 0 is equivalent to q(x) = 0. Using (4.2), we now

see that (3.1) is equivalent to q(b) = 0.Next, it follows from (3.2) that

q′(x) = Ψ(x)(L − r)hs(x)

> 0 if x < xs,

< 0 if x > xs.(4.11)

This, together with the fact that limx→−∞ q(x) = 0, implies that there exists a unique b∗ suchthat q(b∗) = 0 if and only if limx→+∞ q(x) < 0. Next, we show that this inequality holds. By thedefinition of hs and F , we have

hs(x)

F (x)=ex − csF (x)

> 0 for x > ln cs, limx→+∞

hs(x)

F (x)= 0,

(

hsF

)′

(x) =W(x)

F 2(x)

∫ x

−∞Ψ(s)(L − r)hs(s)ds =

W(x)

F 2(x)q(x). (4.12)

Since q is strictly decreasing in (xs,+∞), the above hold true if and only if limx→+∞ q(x) < 0.Therefore, we conclude that there exits a unique b∗ such that ebF (b) = (eb − cs)F

′(b). Using (4.2),we see that

b∗ > xs and q(x) > 0 for all x < b∗. (4.13)

Observing that eb∗, F (b∗), F ′(b∗) > 0, we can conclude that hs(b

∗) = eb∗− cs > 0, or equivalently

b∗ > ln cs.We now verify by direct substitution that V (x) and J(x) in (3.3) satisfy the pair of variational

inequalities:

minrJ(x)− LJ(x), J (x)− (V (x)− hb(x)) = 0, (4.14)

minrV (x)− LV (x), V (x)− (J(x) + hs(x)) = 0. (4.15)

First, note that J(x) is identically 0 and thus satisfies the equality

(r − L)J(x) = 0. (4.16)

To show that J(x) − (V (x) − hb(x)) ≥ 0, we look at the disjoint intervals (−∞, b∗) and [b∗,∞)separately. For x ≥ b∗, we have

V (x)− hb(x) = −(cb + cs),

which implies J(x)− (V (x)− hb(x)) = cb + cs ≥ 0. When x < b∗, the inequality

J(x)− (V (x)− hb(x)) ≥ 0

can be rewritten as

hb(x)

F (x)=ex + cbF (x)

≥eb

∗− cs

F (b∗)=hs(b

∗)

F (b∗). (4.17)

To determine the necessary conditions for this to hold, we consider the derivative of the LHS of(4.2):

(

hbF

)′

(x) =W(x)

F 2(x)

∫ x

−∞Ψ(s)(L − r)hb(s)ds =

W(x)

F 2(x)

∫ x

−∞Ψ(s)esfb(s)ds. (4.18)

16

If fb(x) = 0 has no roots, then (L − r)hb(x) is negative for all x ∈ R. On the other hand, if thereis only one root x, then (L − r)hb(x) = 0 and (L − r)hb(x) < 0 for all other x. In either case,hb(x)/F (x) is a strictly decreasing function and (4.2) is true.

Otherwise if fb(x) = 0 has two distinct roots xb1 and xb2 with xb1 < xb2, then

(L − r)hb(x)

< 0 if x ∈ (−∞, xb1) ∪ (xb2,+∞),

> 0 if x ∈ (xb1, xb2).(4.19)

Applying (4.2) to (4.2), the derivative (hb/F )′(x) is negative on (−∞, xb1). Hence, hb(x)/F (x) is

strictly decreasing on (−∞, xb1). We further note that b∗>xs>xb2. Observe that on the interval(xb1, xb2), the intergrand is positive. It is therefore possible for (hb/F )

′ to change sign at somex ∈ (xb1, xb2). For this to happen, the positive part of the integral must be larger than the absolutevalue of the negative part. In other words, (3.2) must hold. If (3.2) holds, then there must existsome a∗ ∈ (xb1, xb2) such that (hb/F )

′(a∗) = 0, or equivalently (ii) holds:

(

hbF

)′

(a∗) =h′b(a

∗)

F (a∗)−hb(a

∗)F ′(a∗)

F 2(a∗)=

ea∗

F (a∗)−

(ea∗+ cb)F (a

∗)′

F 2(a∗).

If (ii) holds, then we have

∣

∣

∣

∣

∫ xb1

−∞Ψ(x)exfb(x)dx

∣

∣

∣

∣

=

∫ a∗

xb1

Ψ(x)exfb(x)dx.

In addition, since

∫ xb2

a∗Ψ(x)exfb(x)dx > 0,

it follows that∣

∣

∣

∣

∫ xb1

−∞Ψ(x)exfb(x)dx

∣

∣

∣

∣

<

∫ xb2

xb1

Ψ(x)exfb(x)dx.

This establishes the equivalence between (ii) and (3.2). Under this condition, hb/F is strictlydecreasing on (xb1, a

∗). Then, either it is strictly increasing on (a∗, b∗), or there exists some x ∈(xb2, b

∗) such that hb(x)/F (x) is strictly increasing on (a∗, x) and strictly decreasing on (x, b∗). Inboth cases, (4.2) is true if and only if (ii) holds.

Alternatively, if (3.2) doesn’t hold, then by in (4.2), the integral (hb/F )′ will always be negative.

This means that the function hb(x)/F (x) is strictly decreasing for all x ∈ (−∞, b∗), in which case(4.2) holds.

We are thus able to show that (4.2) holds, in particular the minimum of 0 is achieved as a resultof (4.2). To prove (4.2), we go through a similar procedure. To check that

(r − L)V (x) ≥ 0

holds, we consider two cases. First when x < b∗, we get

(r − L)V (x) =eb

∗− cs

F (b∗)(r − L)F (x) = 0.

On the other hand, when x ≥ b∗, the inequality holds

(r − L)V (x) = (r − L)hs(x) > 0,

17

since b∗ > xs (the first inequality of (4.2)) and (3.2).Similarly, when x ≥ b∗, we have

V (x)− (J(x) + hs(x)) = hs(x)− hs(x) = 0.

When x < b∗, the inequality holds:

V (x)− (J(x) + hs(x)) =hs(b

∗)

F (b∗)F (x)− hs(x) ≥ 0,

which is equivalent to hs(x)F (x) ≤ hs(b∗)

F (b∗) , due to (4.2) and (4.2).

Proof of Theorem 3.4 (Part 1) Define the functions

qG(x, z) =

∫ +∞

x

Φ(s)(L − r)hb(s)ds −

∫ +∞

z

Φ(s)(L − r)hs(s)ds,

qF (x, z) =

∫ x

−∞Ψ(s)(L − r)hb(s)ds−

∫ z

−∞Ψ(s)(L − r)hs(s)ds.

We look for the points d∗ < b∗ such that

qG(d∗, b∗) = 0, and qF (d

∗, b∗) = 0.

This is because these two equations are equivalent to (3.4) and (3.4), respectively.Now we start to solve the equations by first narrowing down the range for d∗ and b∗. Observe

that

qG(x, z) =

∫ z

x

Φ(s)(L − r)hb(s)ds+

∫ ∞

z

Φ(s)[(L − r)(hb(s)− hs(s)]ds

=

∫ z

x

Φ(s)(L − r)hb(s)ds− r(cb + cs)

∫ ∞

z

Φ(s)ds

< 0, (4.20)

for all x and z such that xb2 ≤ x < z. Therefore, d∗ ∈ (−∞, xb2).Since b∗ > xs satisfies q(b

∗) = 0 and a∗ < xb2 satisfies (ii), we have

limz→+∞

qF (x, z) =

∫ x

−∞Ψ(s)(L − r)hb(s)ds− q(b∗)−

∫ +∞

b∗Ψ(s)(L − r)hs(s)ds > 0,

for all x ∈ (a∗, xb2). Also, we note that

∂qF∂z

(x, z) = −Ψ(z)(L − r)hs(z)

< 0 if z < xs,

> 0 if z > xs,(4.21)

and

qF (x, x) =

∫ x

−∞Ψ(s)(L − r) [hb(s)− hs(s)] ds = −r(cb + cs)

∫ x

−∞Ψ(s)ds < 0. (4.22)

Then, (4.2) and (4.2) imply that there exists a unique function β : [a∗, xb2) 7→ R s.t. β(x) > xs and

qF (x, β(x)) = 0. (4.23)

18

Differentiating (4.2) with respect to x, we see that

β′(x) =Ψ(x)(L − r)hb(x)

Ψ(β(x))(L − r)hs(β(x))< 0,

for all x ∈ (xb1, xb2). In addition, by the facts that b∗ > xs satisfies q(b∗) = 0, a∗ satisfies (ii), andthe definition of qF , we have

β(a∗) = b∗.

By (4.2), we have limx↑xb2 qG(x, β(x)) < 0. By computation, we get that

d

dxqG(x, β(x)) = −

Φ(x)Ψ(β(x))− Φ(β(x))Ψ(x)

Ψ(β(x))(L − r)hb(x)

= −Ψ(x)

[

G(x)

F (x)−G(β(x))

F (β(x))

]

(L − r)hb(x) < 0,

for all x ∈ (xb1, xb2). Therefore, there exists a unique d∗ such that qG(d∗, β(d∗)) = 0 if and only if

qG(a∗, β(a∗)) > 0.

The above inequality holds if (3.4) holds. Indeed, direct computation yields the equivalence:

qG(a∗, β(a∗)) =

∫ +∞

a∗Φ(s)(L − r)hb(s)ds−

∫ +∞

b∗Φ(s)(L − r)hs(s)ds

= −hb(a

∗)

F (a∗)−G(b∗)

F (b∗)

∫ b∗

−∞Ψ(s)(L − r)hs(s)ds −

∫ +∞

b∗Φ(s)(L − r)hs(s)ds

= −ea

∗+ cb

F (a∗)+eb

∗− cs

F (b∗).

When this solution exists, we have

d∗ ∈ (xb1, xb2) and b∗ := β(d∗) > xs.

Next, we show that the functions J and V given in (3.4) and (3.4) satisfy the pair of VIs in (4.2)and (4.2). In the same vein as the proof for the Theorem 3.3, we show

(r −L)J(x) ≥ 0

by examining the 3 disjoint regions on which J(x) assume different forms. When x < a∗,

(r − L)J(x) = P (r − L)F (x) = 0.

Next, when x > d∗,

(r − L)J(x) = Q(r − L)G(x) = 0.

Finally for x ∈ [a∗, d∗],

(r − L)J(x) = (r − L)(KF (x)− hb(x)) = −(r − L)hb(x) > 0,

as a result of (4.2) since a∗, d∗ ∈ (xb1, xb2).

19

Next, we verify that

(r − L)V (x) ≥ 0.

Indeed, we have (r − L)V (x) = K(r − L)F (x) = 0 for x < b∗. When x ≥ b∗, we get the inequality(r − L)V (x) = (r − L)(QG(x) + hs(x)) = (r − L)hs(x) > 0 since b∗ > xs and due to (3.2).

It remains to show that J(x)− (V (x)−hb(x)) ≥ 0 and V (x)− (J(x)+hs(x)) ≥ 0. When x < a∗,we have

J(x)− (V (x)− hb(x)) = (P − K)F (x) + (ex + cb) = −F (x)ea

∗+ cb

F (a∗)+ (ex + cb) ≥ 0.

This inequality holds since we have shown in the proof of Theorem 3.3 that hb(x)F (x) is strictly decreasing

for x < a∗. In addition,

V (x)− (J(x) + hs(x)) = F (x)ea

∗+ cb

F (a∗)− (ex − cs) ≥ 0,

since (4.2) (along with the ensuing explanation) implies that hs(x)F (x) is increasing for all x < a∗.

In the other region where x ∈ [a∗, d∗], we have

J(x)− (V (x)− hb(x)) = 0,

V (x)− (J(x) + hs(x)) = hb(x)− hs(x) = cb + cs ≥ 0.

When x > b∗, it is clear that

J(x)− (V (x)− hb(x)) = hb(x)− hs(x) = cb + cs ≥ 0,

V (x)− (J(x) + hs(x)) = 0.

To establish the inequalities for x ∈ (d∗, b∗), we first denote

gJ(x) := J(x)− (V (x)− hb(x)) = QG(x)− KF (x) + hb(x)

= F (x)

∫ x

d∗Φ(s)(L − r)hb(s)ds−G(x)

∫ x

d∗Ψ(s)(L − r)hb(s)ds,

gV (x) := V (x)− (J(x) + hs(x)) = KF (x)− QG(x)− hs(x)

= F (x)

∫ b∗

x

Φ(s)(L − r)hs(s)ds−G(x)

∫ b∗

x

Ψ(s)(L − r)hs(s)ds.

In turn, we compute to get

g′J(x) = F ′(x)

∫ x

d∗Φ(s)(L − r)hb(s)ds−G′(x)

∫ x

d∗Ψ(s)(L − r)hb(s)ds,

g′V(x) = F ′(x)

∫ b∗

x

Φ(s)(L − r)hs(s)ds−G′(x)

∫ b∗

x

Ψ(s)(L − r)hs(s)ds.

Recall the definition of xb2 and xs, and the fact that G′ < 0 < F ′, we have g′J(x) > 0 for x ∈ (d∗, xb2)

and g′V(x) < 0 for x ∈ (xs, b

∗). These, together with the fact that gJ(d∗) = gV (b

∗) = 0, imply that

gJ (x) > 0 for x ∈ (d∗, xb2), and gV (x) > 0 for x ∈ (xs, b∗).

20

Furthermore, since we have

gJ(b∗) = cb + cs ≥ 0, gV (d

∗) = cb + cs ≥ 0, (4.24)

and

(L − r)gJ(x) = (L − r)hb(x) < 0 for all x ∈ (xb2, b∗), (4.25)

(L − r)gV (x) = −(L − r)hs(x) < 0 for all x ∈ (d∗, xs).

In view of inequalities (4.2) and (4.2), the maximum principle implies that gJ (x) ≥ 0 and gV (x) ≥ 0

for all x ∈ (d∗, b∗). Hence, we conclude that J(x)−(V (x)−hb(x)) ≥ 0 and V (x)−(J(x)+hs(x)) ≥ 0hold for x ∈ (d∗, b∗).

Proof of Theorems 3.3 and 3.4 (Part 2) We now show that the candidate solutions in Theo-rems 3.3 and 3.4, denoted by j and v, are equal to the optimal switching value functions J and Vin (2.2) and (2.2), respectively. First, we note that j ≤ J and v ≤ V , since J and V dominate theexpected discounted cash low from any admissible strategy.

Next, we show the reverse inequaities. In Part 1, we have proved that j and v satisfy the VIs(4.2) and (4.2). In particular, we know that (r − L)j ≥ 0, and (r − L)v ≥ 0. Then by Dynkin’sformula and Fatou’s lemma, as in Øksendal (2003, p. 226), for any stopping times ζ1 and ζ2 suchthat 0 ≤ ζ1 ≤ ζ2 almost surely, we have the inequalities

Exe−rζ1 j(Xζ1) ≥ Exe

−rζ2 j(Xζ2), and Exe−rζ1 v(Xζ1) ≥ Exe

−rζ2 v(Xζ2). (4.26)

For Λ0 = (ν1, τ1, ν2, τ2, . . . ), noting that ν1 ≤ τ1 almost surely, we have

j(x) ≥ Exe−rν1 j(Xν1) (4.27)

≥ Exe−rν1(v(Xν1)− hb(Xν1)) (4.28)

≥ Exe−rτ1 v(Xτ1) − Exe

−rν1hb(Xν1) (4.29)

≥ Exe−rτ1(j(Xτ1) + hs(Xτ1)) − Exe

−rν1hb(Xν1) (4.30)

= Exe−rτ1 j(Xτ1)+ Exe

−rτ1hs(Xτ1)− e−rν1hb(Xν1), (4.31)

where (4.2) and (4.2) follow from (4.2). Also, (4.2) and (4.2) follow from (4.2) and (4.2) respectively.Observing that (4.2) is a recursion and j(x) ≥ 0 in both Theorems 3.3 and 3.4, we obtain

j(x) ≥ Ex

∞∑

n=1

[e−rτnhs(Xτn)− e−rνnhb(Xνn)]

.

Maximizing over all Λ0 yields that j(x) ≥ J(x). A similar proof gives v(x) ≥ V (x).

Remark 4.4 If there is no transaction cost for entry, i.e. cb = 0, then fb, which is now a linearfunction with a non-zero slope, has one root x0. Moreover, we have fb(x) > 0 for x ∈ (−∞, x0) andfb(x) < 0 for x ∈ (x0,+∞). This implies that the entry region must be of the form (−∞, d0), forsome number d0. Hence, the continuation region for entry is the connected interval (d0,∞).

Remark 4.5 Let Lξ be the infinitesimal generator of the XOU process ξ = eX , and define thefunction Hb(y) := y + cb ≡ hb(ln y). In other words, we have the equivalence:

(Lξ − r)Hb(y) ≡ (L − r)hb(ln y).

21

Referring to (3.2) and (3.2), we have either that

(Lξ − r)Hb(y)

> 0 for y ∈ (yb1, yb2),

< 0 for y ∈ (0, yb1) ∪ (yb2,∞),(4.32)

where yb1 = exb1 > 0 and yb2 = exb2 and xb1 < xb2 are two distinct roots to (3.2), or

(Lξ − r)Hb(y) < 0, for y ∈ (0, y∗) ∪ (y∗,∞), (4.33)

where y∗ = ex∗and x∗ is the single root to (3.2). In both cases, Assumption 4 of Zervos et al.

(2013) is violated, and their results cannot be applied. Indeed, they would require that (Lξ− r)Hb(y)is strictly negative over a connected interval of the form (y0,∞), for some fixed y0 ≥ 0. However, itis clear from (4.5) and (4.5) that such a region is disconnected.

In fact, the approach by Zervos et al. (2013) applies to the optimal switching problems wherethe optimal wait-for-entry region (in log-price) is of the form (d∗,∞), rather than the disconnectedregion (−∞, a∗)∪ (d∗,∞), as in our case with an XOU underlying. Using the new inferred structureof the wait-for-entry region, we have modified the arguments in Zervos et al. (2013) to solve ouroptimal switching problem for Theorems 3.3 and 3.4.

A Appendix

A.1 Proof of Lemma 4.2 (Properties of H). The continuity and twice differentiability ofH on (0,+∞) follow directly from those of hs, G and ψ. On the other hand, we have H(0) :=

limx→−∞

(hs(x))+

G(x) = limx→−∞

(ex−cs)+

G(x) = limx→−∞

0G(x) = 0. Hence, the continuity of H at 0 follows from

limz→0

H(z) = limx→−∞

hs(x)

G(x)= lim

x→−∞

ex − csG(x)

= 0.

Next, we prove properties (i)-(iii) of H.(i) This follows trivially from the fact that ψ(x) is a strictly increasing function and G(x) > 0.(ii) By the definition of H,

H ′(z) =1

ψ′(x)(hsG)′(x) =

[exG(x)− (ex − cs)G′(x)]

ψ′(x)G2(x), z = ψ(x).

For x ∈ (ln cs,+∞), ex − cs > 0, G′(x) < 0, so exG(x)− (ex − cs)G′(x) > 0. Also, since both ψ′(x)

and G2(x) are positive, we conclude that H ′(z) > 0 for z ∈ (ψ(ln cs),+∞).The proof of the limit of H ′(z) will make use of property (iii), and is thus deferred until after

the proof of property (iii).(iii) By differentiation, we have

H ′′(z) =2

σ2G(x)(ψ′(x))2[(L − r)hs](x), z = ψ(x).

Since σ2, G(x) and (ψ′(x))2 are all positive, we only need to determine the sign of (L − r)hs(x) =exfs(x). Hence, property (iii) follows from (3.2).

To find the limit of H ′(z), we first observe that

limx→+∞

hs(x)

F (x)= 0. (A.1)

22

Indeed, we have

limx→+∞

hs(x)

F (x)= lim

x→+∞

1

e−xF (x)= lim

x→+∞

(∫ +∞

0u

rµ−1e(√

2µ

σ2−1u)xu−

√

2µ

σ2 θu−u2

2 du

)−1

= limx→+∞

∫

√

σ2

2µ

0u

rµ−1e(√

2µ

σ2−1u)xu−

√

2µ

σ2 θu−u2

2 du+

∫ +∞

√

σ2

2µ

urµ−1e(√

2µ

σ2−1u)xu−

√

2µ

σ2 θu−u2

2 du

−1

.

Since the first term on the RHS is non-negative and the second term is strictly increasing and convexin x, the limit is zero.

Turning now to H ′(z), we note that

H ′(z) =1

ψ′(x)(hsG

)′(x), z = ψ(x).

As we have shown, for z > ψ(ln cs) ∧ ψ(xs), H′(z) is a positive and decreasing function. Hence the

limit exists and satisfies

limz→+∞

H ′(z) = limx→+∞

1

ψ′(x)(hsG)′(x) = c ≥ 0. (A.2)

Observe that limx→+∞hs(x)G(x) = +∞, limx→+∞ ψ(x) = +∞, and limx→+∞

(hs(x)G(x)

)′

ψ′(x) exists, and ψ′(x) 6=0. We apply L’Hopital’s rule to get

limx→+∞

hs(x)

F (x)= lim

x→+∞

hs(x)G(x)

F (x)G(x)

= limx→+∞

(hs(x)G(x) )

′

ψ′(x)= c. (A.3)

Comparing (A) and (A) implies that c = 0. From (A), we conclude that limz→+∞H ′(z) = 0.

A.2 Proof of Lemma 4.3 (Properties of H). It is straightforward to check that V (x) iscontinuous and differentiable everywhere, and twice differentiable everywhere except at x = b∗. Thesame properties hold for h(x). Since both G and ψ are twice differentiable everywhere, the continuityand differentiability of H on (0,+∞) and twice differentiability on (0, ψ(b∗)) ∪ (ψ(b∗),+∞) followdirectly.

To see the continuity of H(y) at 0, note that V (x) → 0 and ex → 0 as x→ −∞. Then we have

H(0) := limx→−∞

(h(x))+

G(x)= lim

x→−∞

(V (x)− ex − cb)+

G(x)= lim

x→−∞

0

G(x)= 0,

and limz→0 H(z) = limx→−∞h(x)G(x) = limx→−∞

−cbG(x) = 0. There follows the continuity at 0.

(i) For x ∈ [b∗,+∞), we have h(x) ≡ −(cs + cb) < 0 . Next, the limits limx→−∞

V (x) → 0 and

limx→−∞

ex → 0 imply that limx→−∞

h(x) = V (x) − ex − cb → −cb < 0. Therefore, there exists some b

such that h(x) < 0 for x ∈ (−∞, b). For the non-trivial case in question, h(x) must be positive forsome x, so we must have b < b∗. To conclude, we have h(x) < 0 for x ∈ (−∞, b) ∪ [b∗,+∞). This,along with the facts that ψ(x) ∈ (0,+∞) is a strictly increasing function and G(x) > 0, impliesproperty (i).(ii) By differentiating H(z), we get

H′

(z) =1

ψ′(x)(h

G)′(x), z = ψ(x).

23

To determine the sign of H′, we observe that, for x ≥ b∗,

(h(x)

G(x))′ = (

−(cs + cb)

G(x))′ =

(cs + cb)G′(x)

G2(x)< 0.

Also, ψ′(x) > 0 for x ∈ R. Therefore, H(z) is strictly decreasing for z ≥ ψ(b∗).(iii) To study the convexity/concavity, we look at the second derivative

H′′

(z) =2

σ2G(x)(ψ′(x))2(L − r)h(x), z = ψ(x).

Since σ2, G(x) and (ψ′(x))2 are all positive, we only need to determine the sign of (L − r)h(x):

(L − r)h(x) =σ2

2(V ′′(x)− ex) + µ(θ − x)(V ′(x)− ex)− r(V (x)− ex − cb)

=

[µx− (µθ + σ2

2 − r)]ex + rcb if x ∈ (−∞, b∗),

r(cs + cb) > 0 if x ∈ (b∗,+∞).

which suggests that H(z) is convex for z ∈ (ψ(b∗),+∞).Furthermore, for x ∈ (xs, b

∗), we have

(L − r)h(x) = [µx− (µθ +σ2

2− r)]ex + rcb = −exfs(x) + r(cs + cb) > r(cs + cb) > 0,

by the definition of xs. Therefore, H(z) is also convex on (ψ(xs), ψ(b∗)). Thus far, we have estab-

lished that H(z) is convex on (ψ(xs),+∞).Next, we determine the convexity of H(z) on (0, ψ(xs)]. Denote z1 := argmaxz∈[0,+∞) H(z).

Since supx∈R h(x) > 0, we must have H(z1) = supz∈[0,+∞) H(z) > 0. By its continuity and differ-

entiability, H must be concave at z1. Then, there must exist some interval (ψ(a(0)), ψ(d(0))) overwhich H is concave and z1 ∈ (ψ(a(0)), ψ(d(0))).

On the other hand, for x ∈ (−∞, xs],

((L − r)h)′(x) = [µx− (µθ +σ2

2− r − µ)]ex

< 0 if x ∈ (−∞, x∗),

> 0 if x ∈ (x∗, xs],

where x∗ = θ + σ2

2µ − rµ− 1. Therefore, (L − r)h(x) is strictly decreasing on (−∞, x∗), strictly

increasing on (x∗, xs], and is strictly positive at xs and −∞:

(L − r)h(xs) = r(cs + cb) > 0 and limx→−∞

(L − r)h(x) = rcb > 0.

If (L − r)h(x∗) = −µex∗+ rcb < 0, then there exist exactly two distinct roots to the equation

(L − r)h(x) = 0, denoted as xb1 and xb2, such that −∞ < xb1 < x∗ < xb2 < xs and

(L − r)h(x)

> 0 if x ∈ (−∞, xb1) ∪ (xb2, xs],

< 0 if x ∈ (xb1, xb2).

On the other hand, if (L − r)h(x∗) = −µex∗+ rcb ≥ 0, then (L − r)h(x) ≥ 0 for all x ∈ R, and

H(z) is convex for all z, which contradicts with the existence of a concave interval. Hence, weconclude that −µex

∗+ rcb < 0, and (xb1, xb2) is the unique interval that (L − r)h(x) < 0. Con-

sequently, (a(0), d(0)) coincides with (xb1, xb2) and z1 ∈ (ψ(xb1), ψ(xb2)). This completes the proof.

24

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